Complex numbers practice · Medium

The correct answer is A ($x^{{2}} - 4x + 1 = 0$) because it is the quadratic equation whose roots are derived from the given complex number equation.

Option C ($\arg \left( {z - 1} \right) = \mathrm{\alpha }$) is correct because the imaginary part of the expression is zero when $\arg(z-1) = \alpha$.

Option D ($- i$) is correct because it is the simplified value of the given summation after applying the formula for the sum of a geometric series and using properties of complex numbers.

Option A ((0,2 tan^-1 1/4]) is correct because it is the interval for 'a' where the discriminant of the quadratic equation is greater than zero.

Option A ($\frac{1}{2}{\left| z \right|^2}$) is correct because the area of the triangle formed by the given vertices is half the square of the magnitude of z.

Option B ($\cos n\phi$) is correct because the expression simplifies to $\cos n\phi + i\sin n\phi$, and thus u = $\cos n\phi$.

Option D ($1 + i\left( {1 \pm \sqrt 3 } \right)$) is correct because it represents the vertices adjacent to $2+i$ in a regular hexagon centered at $i$.

Option A (-2) is correct because by cubing $z^{\frac{1}{3}} = p + iq$ and equating real and imaginary parts, the expression simplifies to -2.

Option C (${x^2} + {y^2} - 2y = 1$) is correct because by substituting $z = x + iy$ into the given argument equation and simplifying, we obtain the equation of a circle.

Option A (0) is correct because the sum of the arguments of a complex number and its conjugate is zero, and z2 and z4 are conjugates of z1 and z3 respectively.

Option B ($-2 i^{n+1}$) is correct because after simplifying the expression using properties of complex numbers, the result matches this option.

Option A ($\frac{1}{2} \arg z$) is correct because after simplification using trigonometric identities, the argument of the expression reduces to half of the argument of z.

Option C (1) is correct because the minimum value of the expression is 1 when a, b, and c are distinct integers.

The correct answer is D ($- 1$) because, in an equilateral triangle, the vertices are equally spaced around the circumscribing circle, and using the properties of complex numbers, z2 = -1.

Option C (-5π/12) is correct because it is the principal argument obtained after calculating the argument of the complex number and adjusting to the range (-π, π].

Option C ($i\cot \frac{\mathrm{\theta }}{2}$) is correct because it is the simplified form of the given expression using Euler's formula and trigonometric identities.

Option D (None of these) is correct because the summation simplifies to 'i', which is not among the given options.

Option A (-$\frac{{\theta}}{{2}}$, $32 \cos ^5 \frac{{\theta}}{{2}}$) is correct because after simplifying using trigonometric identities and De Moivre's theorem, the argument and modulus are -$\frac{{\theta}}{{2}}$ and $32 \cos ^5 \frac{{\theta}}{{2}}$, respectively.

Option B (Rational number) is correct because given the conditions, |z⁽²ⁿ⁾ - 1| can be shown to be a rational number using trigonometric identities and properties of rational numbers.

Option A (A circle) is correct because the given equation can be rearranged into the standard form of a circle's equation in the complex plane.

Option C ($3 \cdot 4^{2n}, n$ is multiple of 3) is correct because when n is a multiple of 3, the expression simplifies to this value.

Option C (1) is correct because after simplification using Euler's formula, the expression evaluates to 1.

Option C (-i) is correct because by converting to exponential form and using geometric series formula, the sum simplifies to -i.

The correct answer is A (0) because the given equation simplifies to |z-2| = |z+2|, which represents the perpendicular bisector of the line joining 2 and -2, i.e., the y-axis. The minimum value of |z| is 0, occurring at the origin.

Option D (None of these) is correct because the multiplicative inverse of $(6+5i)^2$ is $\frac{11}{(61)^2} - \frac{60}{(61)^2}i$, which is not among the given options.

Option C (6,0) is correct because the maximum and minimum values of $|z+1|$ are found by using the triangle inequality on $|(z+4)-3|$.

Option C (1) is correct because the absolute value of the nth term of the geometric progression is 1, as the magnitude of the common ratio is 1.

Option B ($- 160$) is correct because substituting $x = - 5 + 2\sqrt { - 4}$ into the expression ${x^4} + 9{x^3} + 35{x^2} - x + 4$ and simplifying results in -160.

Option C ($\frac{{\|u|-|v\|}}{{1+|u||v|}}$) is correct because by using properties of complex numbers and the reverse triangle inequality, we can establish the lower bound for |z|.

Option A (8,2) is correct because the maximum and minimum values of $|z+3|$ are found by using the triangle inequality on $|(z+4i)+(3-4i)|$.